To find one factor of the expression [tex]\((x+1)^3 - (x+1)\)[/tex], we will follow these steps:
1. Simplify the Expression:
Start with the given expression:
[tex]\[
(x+1)^3 - (x+1)
\][/tex]
2. Factor out Common Terms:
Notice that [tex]\((x+1)\)[/tex] is a common term in both parts of the expression. We can factor [tex]\((x+1)\)[/tex] out:
[tex]\[
(x+1)((x+1)^2 - 1)
\][/tex]
3. Simplify Further Using a Special Formula:
Recognize that [tex]\((x+1)^2 - 1\)[/tex] is a difference of squares, which can be factored as:
[tex]\[
(x+1)^2 - 1 = ((x+1) - 1)((x+1) + 1) = (x)(x+2)
\][/tex]
4. Combine Steps:
Substitute back into the factored form:
[tex]\[
(x+1)((x)(x+2))
\][/tex]
This can be further simplified to:
[tex]\[
(x+1) \cdot x \cdot (x+2)
\][/tex]
Bringing everything together, the factored form of the expression is:
[tex]\[
(x+1) \cdot x \cdot (x+2)
\][/tex]
From this, one can extract that the expression has been factored into three parts: [tex]\((x+1)\)[/tex], [tex]\(x\)[/tex], and [tex]\((x+2)\)[/tex].
Therefore, one of the factors of [tex]\((x+1)^3 - (x+1)\)[/tex] is:
[tex]\[
x
\][/tex]
